Temperatures of a Thin Rod. 233 



that some characteristic fraction of about the same order of 

 magnitude holds for a square bar), we have the first ratio 

 about 1/100. Despretz also employed the method to deter- 

 mine the conductivity of marble, porcelain, and other non- 

 conductors, but got completely discordant results. The 

 fraction (i'i + r 3 )/r 2 , which should be constant for the same 

 bar, varied from 10'83 to 3'87 for marble. From the table 

 we see that the characteristic ratio for marble is for the size 

 of bar employed by Despretz about 1/30, a value much too 

 large for the solution to be applicable. 



The experiments of Wiedemann and Franz* illustrate the 

 need for the second criterion. They used much thinner bars 

 (0*4 to 0*6 cm. in diameter), and the observations were con- 

 fined to metals only, so that for both these reasons the first 

 condition is satisfied. But no attempt seems to have been 

 made to choose appropriate lengths for the different sub- 

 stances. All the bars were made 50 cms. long, so that the 

 second condition is not satisfied. In the account of the ex- 

 periments they state that the expression (i^ + v^lv? does not, 

 for the highly conducting substances, vary appreciably from 

 2, and that therefore the values of the conductivity for these 

 substances are not very reliable. This value 2 implies that 

 the curve of temperatures is not logarithmic but rectilinear, 

 the arithmetical progression has taken the place of the geo- 

 metrical progression ; in other words, the lateral radiation is 

 non-effective, and the end radiation is effective in producing 

 the temperature drop, so that the conditions of the problem 

 are not satisfied. If we work out for the length employed 

 (50 cms.) and for the case of copper (taking the larger value 

 of. the thermal length modulus since in these experiments 

 the bars were plated) we find the characteristic fraction 



\J ~y '• I to be about 1/2, L e. the condition is not satisfied. 



The same holds true in a worse degree for the silver bars 

 employed in these experiments. 



Of course this method is not now regarded as satisfactory 

 except for the highly conducting "substances, and in several 

 places f w*e find it stated that the conductivity should be 

 large and the cross-section of the bar small. But so far as 1 

 can find these two relationships of the four lengths are not 

 given. To say that k is to be large and a small gives an 

 approach to the conditions ; but to get an exact idea of the 



* Ami. de Chimie et de Physique, t. xli. p. 107 (1854). 

 t Kelvin. Article " Heat,'' in Encyc. Britann. §78: Tait, Text -book- 

 on Heat, p. 213. 



